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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorAndrew Stacey
   • CommentTimeMar 18th 2014
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexHi, I happened to notice today that there are problems with the proof of Lemma 1. Unfortunately, I do not have time to fix this myself, but I thought I'd let you know. It can be fixed as follows. 1) Define $\hat{X}$ to be the pullback of $f \times id : X \times Y \rightarrow Y \times Y$ and the map $Y^{I} \rightarrow Y \times Y$ which is currently denoted (I would not use this notation myself!) by $(d_0,d_1)$. 2) The required map $p : Z \rightarrow Y$ is the composite of the map $Z \rightarrow X \times Y$ which is part of the pullback of 1), and the projection map $X \times Y \rightarrow Y$. 3) The required fibration $Z \rightarrow X$ arises in the same way as in 2), but composing with the the other projection map instead, 4) The required map $X \rightarrow Z$ arises via the universal property of the pullback by using the map $f \circ c : X \rightarrow Y^I$, where $c$ is the map $Y \rightarrow Y^I$ appearing in the factorisation which defines $Y^I$, and the map $id \times f : X \rightarrow X \times Y$.

   Hi,

   I happened to notice today that there are problems with the proof of Lemma 1. Unfortunately, I do not have time to fix this myself, but I thought I’d let you know.

   It can be fixed as follows.

   1) Define to be the pullback of and the map which is currently denoted (I would not use this notation myself!) by .

   2) The required map is the composite of the map which is part of the pullback of 1), and the projection map .

   3) The required fibration arises in the same way as in 2), but composing with the the other projection map instead,

   4) The required map arises via the universal property of the pullback by using the map , where is the map appearing in the factorisation which defines , and the map .

2. • CommentRowNumber2.
   • CommentAuthorAndrew Stacey
   • CommentTimeMar 18th 2014
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexOkay, so Chrome messes up the preview but is fine once the post is posted. Probably due to the interaction of javascripts between inserting the preview and re-running MathJaX on the page.

   Okay, so Chrome messes up the preview but is fine once the post is posted. Probably due to the interaction of javascripts between inserting the preview and re-running MathJaX on the page.

3. • CommentRowNumber3.
   • CommentAuthorAndrew Stacey
   • CommentTimeMar 18th 2014
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexFor my own future reference, the solution would appear to be [documented here](http://docs.mathjax.org/en/latest/typeset.html).

   For my own future reference, the solution would appear to be documented here.

4. • CommentRowNumber4.
   • CommentAuthorAndrew Stacey
   • CommentTimeMar 18th 2014
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexAlso, the initial post in this thread is a copy of a post by Richard Williamson who reported the issue (so the "I" is not me but him!).

   Also, the initial post in this thread is a copy of a post by Richard Williamson who reported the issue (so the “I” is not me but him!).